Applications of algorithmic differentiation to phase retrieval algorithms

Alden S. Jurling; James R. Fienup

doi:10.1364/JOSAA.31.001348

Journal of the Optical Society of America A
Vol. 31,
Issue 7,
pp. 1348-1359
(2014)
•https://doi.org/10.1364/JOSAA.31.001348

Applications of algorithmic differentiation to phase retrieval algorithms

Alden S. Jurling and James R. Fienup

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Alden S. Jurling and James R. Fienup, "Applications of algorithmic differentiation to phase retrieval algorithms," J. Opt. Soc. Am. A 31, 1348-1359 (2014)

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Related Topics
Table of Contents Category
- Image Processing
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Wave-front sensing (010.7350)
- Mathematical methods in physics (000.3860)
- Numerical approximation and analysis (000.4430)
- Phase retrieval (100.5070)

About this Article
History
- Original Manuscript: April 2, 2014
- Revised Manuscript: April 4, 2014
- Manuscript Accepted: April 22, 2014
- Published: June 9, 2014

Abstract

In this paper, we generalize the techniques of reverse-mode algorithmic differentiation to include elementary operations on multidimensional arrays of complex numbers. We explore the application of the algorithmic differentiation to phase retrieval error metrics and show that reverse-mode algorithmic differentiation provides a framework for straightforward calculation of gradients of complicated error metrics without resorting to finite differences or laborious symbolic differentiation.

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Equations (86)

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Name	Forward Model	Gradient Model	Parameter Domains	#
Addition	$z = x + y$	$\bar{x} = \bar{z}$ , $\bar{y} = \bar{z}$	$x, y, z \in C^{N}$	(47)
Scalar–vector addition	$z = x + y$	$\bar{x} = \sum_{n} {\bar{z}}_{n}$ , $\bar{y} = \bar{z}$	$z, y \in C^{N}$ , $x \in C^{1}$	(48)
Element-wise product	$z = x \circ y$	$\bar{x} = y^{} \circ \bar{z}$ , $\bar{y} = x^{} \circ \bar{z}$	$x, y, z \in C^{N}$	(49)
Matrix product	$z = x * y$	$\bar{x} = \bar{z} * y^{T }$ , ${\bar{y} = x}^{T } * \bar{z}$	$x \in C^{N \times P}$ , $y \in C^{P \times M}$ , $z \in C^{N \times M}$	(50)
Scalar–vector product	$z = x y$	$\bar{x} = \sum_{n} y_{n}^{} {\bar{z}}_{n}$ , $\bar{y} = x^{} \bar{z}$	$z, y \in C^{N}$ , $x \in C^{1}$	(51)
Raising to a constant power	$y = x^{c}$	$\bar{x} = c \circ {(x^{*})}^{c - 1} \circ \bar{y}$	$x, y \in C^{N}$ , $c \in R^{N}$	(52)
Element-wise square magnitude	$y = {\| x \|}^{2}$	$\bar{x} = 2 x \circ R {\bar{y}}$	$x \in C^{N}$ , $y \in R^{N}$	(53)
Real part	$y = R {x}$	$R {\bar{x}} = R {\bar{y}}$	$x \in C^{N}$ , $y \in R^{N}$	(54)
Imaginary part	$y = I {x}$	$I {\bar{x}} = R {\bar{y}}$	$x \in C^{N}$ , $y \in R^{N}$	(55)
Complex exponential	$y = e^{x}$	$\bar{x} = \bar{y} \circ y^{*}$	$x, y \in C^{N}$	(56)
Imaginary complex exponential	$y = e^{i x}$	$\bar{x} = I {\bar{y} \circ y^{*}}$	$x \in R^{N}$ , $y \in C^{N}$	(57)
Basis function expansion	$y = \sum_{n} a_{n} B_{n}$	${\bar{a}}_{n} = \sum_{u, v} {\bar{y}}_{u, v} B_{n} (u, v)$	$y \in R^{N \times M}$ , $B \in R^{P \times N \times M}$ , $a \in R^{P}$	(58)
Array indexing	$y = x 〚 α 〛$	$\bar{x} 〚 α 〛 = \bar{y}$	$x, y \in C^{N}$ , $α$ is an index	(59)
Array assignment	$y 〚 α 〛 = x$	$\bar{x} = \bar{y} 〚 α 〛$	$x, y \in C^{N}$ , $α$ is an index	(60)
Scalar broadcasting	$y_{n} = x, \forall n$	$\bar{x} = \sum_{n} {\bar{y}}_{n}$	$y \in C^{N}$ , $x \in C^{1}$	(61)

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Equations (86)

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Journal of the Optical Society of America A

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